Testing equality of spectral density operators for functional linear processes

TL;DR

This paper introduces a new spectral density operator equality test for functional linear processes, using a Hilbert-Schmidt distance measure, with a bootstrap method for improved finite-sample accuracy, validated through simulations and meteorological data.

Contribution

It develops a fully functional $L^2$-type test for spectral density operators, including a novel bootstrap approach with proven asymptotic validity and consistency.

Findings

Bootstrap method improves finite-sample accuracy.

Test has good size and power in simulations.

Application to meteorological data demonstrates practical utility.

Abstract

The problem of testing equality of the entire second order structure of two independent functional linear processes is considered. A fully functional $L^2$-type test is developed which evaluates, over all frequencies, the Hilbert-Schmidt distance between the estimated spectral density operators of the two processes. The asymptotic behavior of the test statistic is investigated and its limiting distribution under the null hypothesis is derived. Furthermore, a novel frequency domain bootstrap method is developed which approximates more accurately the distribution of the test statistic under the null than the large sample Gaussian approximation obtained. Asymptotic validity of the bootstrap procedure is established and consistency of the bootstrap-based test under the alternative is proved. Numerical simulations show that, even for small samples, the bootstrap-based test has very good size and power behavior. An application to meteorological functional time series is also presented.